Static Knowledge vs. Dynamic Argumentation
A Dual Theory Based on Kripke Semantics

As we learn from Newtonian mechanics, motion is relative rather than absolute [15], and quantum mechanics further tells us about wave-particle duality [22]. Nature proves to be highly symmetric and unified. In epistemic logic, agents’ instantaneous knowledge is statically represented via current possible worlds [7], viz. implicitly through the equivalence class of all the indistinguishable worlds at this moment. Whereas in argumentation theory, the first-class citizens are chronological arguments that are dynamically articulated by different agents so as to attack each other [8], explicitly composing a sequential history of a tit-for-tat debate during a period of time.

Following some pioneering attempts to describe argumentation theory with modal logic [11, 12], a number of recent studies continue to explore assorted means of combining these two dimensions, namely agents’ knowledge and argumentation, into one compound Kripke-like model [21, 5, 19]. Nevertheless as far as we can see, those cursory treatments mostly scratch the surface, for under their settings generally speaking, knowledge and argumentation ipso facto reside within two distinct strata of the model which are evaluated in completely separate ways, and so they are totally free to vary independently without any innate relevance. Such kind of ad hoc solution fails to fully conform to our intuition. Actually just from everyday common sense, we can assuredly comprehend that on the one hand, someone (supposedly honest) argues for something, most simply because he knows/believes what he argues for is true; on the other hand even more noticeably, we always infer others’ knowledge/belief based on their arguments and actions, i.e. what they have said and done — essentially speaking, it is exactly the only way through which we may learn about others’ knowledge. After all, in order to find out what others really know, it would help very little even if we were allowed to biologically dissect others’ brains and examine their nerve cells! Therefore of all things, knowledge and argumentation are fundamentally intertwined.

In this paper, we propose a dual theory which simultaneously deals with both static knowledge and dynamic argumentation, from an intrinsic perspective that exquisitely unveils the close mutual interference between these two concepts. Specifically speaking, we shall define two different types of Kripke models, one for epistemic whilst another for argument, and then we shall demonstrate the two-way transformation between the two types. This is to say literally, given some epistemic-type Kripke model, we are able to compute out its corresponding argument-type Kripke model, and vice versa, the epistemic-type Kripke model can be generated from the argument-type Kripke model as well. In order to justify our framework, not only shall we elaborate on the philosophical intuition behind our definitions, but we shall also rigidly prove a duality theorem so that readers will get positively convinced of our approach’s plausibility. Our novel disposal of the relationship between knowledge and argumentation incisively sheds light on both philosophical depth and mathematical complicacy, furthermore, many particular aspects of our method’s superiority will be vividly exhibited as we leisurely walk through pragmatic examples of our logic’s application. At last we shall briefly discuss a few possible expansions and some other relevant topics.

The rest of this paper is organized as follows. Section 2 defines preliminaries of our framework, in particular two types of Kripke models, namely the epistemic one and the argument one. Section 3 illustrates how we can generate the corresponding argument Kripke model given the current epistemic Kripke model. Section 4 subsequently establishes the reverse generation, i.e. from the current argument Kripke model to its corresponding epistemic Kripke model. Since this latter direction of generation may look a bit more complex, in the next Section 5 we at length analyze a truly practical example of a virtual debate between two candidates for the mayor, which significantly affirms our logic’s affluent utility in the real world. Section 6 then is devoted to technical issues and ends up with a duality theorem on the two-way generation between two types of Kripke models, hence formally justifying our dual definition. Section 7 further discusses several relevant expansions and applications of our framework, especially invoking the famous thought experiment of Maxwell’s demon so as to aver that knowledge is a dynamic notion in essence. Section 8 finally concludes this paper and suggests possible directions of future work.

Fix a nonempty set of propositions $P$ and a nonempty set of agents $I$.

The intuitive readings for epistemic formulae are just as usual, e.g., epistemic formula $\mathcal{K}_{i}\varphi$ means “at the current world, agent $i$ knows $\varphi$”. For argument formulae, intuitively, argument formula $i$ means “the current argument is available to agent $i$”, and argument formula $\mathcal{A}_{p}\xi$ means “all the arguments that attack the current argument with respect to proposition $p$ satisfy $\xi$”. Why we stipulate these intuitive readings for argument formulae will gradually become clear to readers as we continue unfolding our entire framework, but for now, if readers cannot help feeling somehow uneasy about our argument formulae or wondering whether they are really useful at all, then readers are strongly advised to take a quick look at the later Subsection 7.1, where we concisely yet convincingly provide further elaboration on our dual syntax, for example what argument formula $\mathcal{A}_{p}\neg i$ intuitively means, and also mention some possible syntactic expansions.

Anyway, the syntactic respect being as such, on the other side, the semantic respect also turns out to be quite standard Kripke models:

An argument Kripke model in Definition 3 is intuitively interpreted as the following:

• •

  For any $U,V\in\mathbb{W}$ and $p\in P$, $(U,V)\in A(p)$ means “argument $V$ attacks argument $U$ with respect to proposition $p$”. Warning: $(U,V)\in A(p)$ means “$V$ attacks $U$”, not “$U$ attacks $V$”.

• •

  For any $U\in\mathbb{W}$ and $i\in I$, $U\in G(i)$ means “argument $U$ is available for agent $i$ to utter out”. As we see, $\mathbb{W}$ includes all epistemically possible arguments, such as “it must be over 7am now, since it is so bright outside” and “it must be before 7am now, since there are few people on the street”. Nonetheless, for any rational agent $i$ at certain fixed world, apparently both arguments cannot be practically accessible to him at the same time, otherwise it would imply absurdity. Therefore, the subset $G(i)\subseteq\mathbb{W}$ denotes all the arguments currently available to agent $i$.

Semantics in Definition 4 is just routine, based on which readers can easily verify that our intuitive readings for both epistemic and argument formulae are indeed appropriate.

Next, in the following Sections 3 and 4, we are going to rigorously define a two-way generation between these dual types of Kripke models, which constitutes the backbone of this paper.

Our first task is to generate an argument Kripke model $\mathfrak{N}_{g}$, given an epistemic Kripke model $\mathfrak{M}=(S,E,F)$ and the current world $s\in S$. Thus more precisely, $\mathfrak{N}_{g}$ depends both on $\mathfrak{M}$ and on $s$ and may be more explicitly denoted as $\mathfrak{N}_{g}(\mathfrak{M},s)$ or $\mathfrak{N}_{g}(s)$, but we shall anyway denote simply as $\mathfrak{N}_{g}$ as long as $\mathfrak{M}$ and $s$ are clear from the context without any possible confusion.

Now that we are provided with all the epistemically possible worlds, what are possible arguments then? In fact ontologically, an argument advocates an opinion about the world, asserting what the world is like or what the world is not like, such as “the earth is getting warmer and warmer” or “unicorn does not exist”. Therefore technically speaking, an argument is essentially a nonempty subset of possible worlds, claiming that our current world lies within this subset. Of course, an empty set stands for a self-contradictory argument, and we can safely ignore such a case under our innocent assumption that all agents are rational.

If arguments are subsets of possible worlds, then how about attack relations between arguments? In fact with respect to any proposition $p$, an argument (i.e., a nonempty subset of possible worlds) must hold one out of three distinct types of attitudes:

1. 1.

   The argument asserts $p$, i.e., all the possible worlds in the subset satisfy $p$.

2. 2.

   The argument asserts $\neg p$, i.e., all the possible worlds in the subset falsify $p$.

3. 3.

   The argument does not tell whether $p$, i.e., some possible worlds in the subset satisfy $p$ while the others falsify $p$.

Hence, several different versions of attack relations might look plausible:

1. (i)

   Weak attack relation: only case 1 and case 2 attack each other.

2. (ii)

   Medium attack relation: case 1 and case 2 attack each other, and both of them attack case 3.

3. (iii)

   Strong attack relation: any two different cases attack each other.

While all the above versions of attack relations may be quite reasonable by one way or another, in this paper, we choose to stick to the medium attack relation. Our intuition is such: different arguments possess different strength. Some arguments are stronger while others are weaker, and generally speaking, stronger arguments attack weaker arguments. The universal set $S$ itself is a trivial argument, as it just says that every possible world is possible and thus essentially argues nothing at all, therefore, it must be one of the weakest arguments and is supposed to be attacked by many other stronger arguments. Assume that with respect to some proposition $p$, the trivial argument $S$ is case 3, (which indeed seems most likely to be,) then we expect it to be attacked by arguments of case 1 and case 2, but not vice versa. The above explanation accounts for our preference for the medium attack relation. Besides, only the medium attack relation is not necessarily symmetric, a crucial feature which undoubtedly makes the medium attack relation both more philosophically profound and more technically interesting.

By the way, we also point out in the later Subsection 7.2 that Dung’s classic argument framework [8] is probably closer to the weak attack relation. Nevertheless for our project here, the weak attack relation looks pitifully too weak: case 3 is just radically ruled out of the game so that much information about the original epistemic Kripke model will get lost.

Finally, we also have to figure out the last part in the generated argument Kripke model $\mathfrak{N}_{g}=(\mathbb{W},A,G)$, namely about the availability function $G$. Here we simply assume every agent is rational and honest, so that his available arguments should not exceed his own knowledge, that is to say, an argument (i.e., a nonempty subset of possible worlds $U\subseteq S$) is available to an agent $i\in I$ if and only if $U$ is a superset of $i$’s current epistemic equivalence class. Putting all the above discussion together, we obtain the following Definition 5:

It is easy to see that $\mathbb{W}\neq\emptyset$ and that $A(p)$ is irreflexive, thus $\mathfrak{N}_{g}$ in Definition 5 is indeed a well-defined argument Kripke model. In order to vividly demonstrate this generating process, we provide a simple example below:

Our second task is to generate an epistemic Kripke model $\mathfrak{M}_{g}$. In parallel with Definition 5 in Section 3, here our start point should be an argument Kripke model $\mathfrak{N}=(\mathbb{W},A,G)$ together with the current argument $U\in\mathbb{W}$. But what does the “current” argument mean? We understand it as the last argument in a realistic argumentation process consisting of a chronological series of successive arguments, which is also ordinarily regarded as “the currently winning argument”, exactly like Dung points out in [8]:

The way humans argue is based on a very simple principle which is summarized succinctly by an old saying: “The one who has the last word laughs best”.

So suppose we are now provided with an argument Kripke model and the current argument, what on earth should be all the epistemically possible worlds? It could look like a merely impossible mission at the very first glance. Nevertheless, considering how we manage to derive argumentation from knowledge in Section 3, conceptually speaking, if our knowledge determines our argumentation, then our argumentation must also reflect our knowledge the other way round. Also from everyday experience, it is actually quite common and natural for us to infer others’ epistemic states based on their utterances, namely, arguments. Therefore, just as nonempty subsets of possible worlds can represent possible arguments, let us in turn try to seek what kinds of subsets of possible arguments can represent possible worlds.

As it is assumed that all agents are rational, roughly speaking, all the possible arguments available to certain agent should form a consistent subset, however not necessarily complete. Instead, very similar to the canonical model method that gets extensively utilized in completeness proof of modal logic [10], the core philosophical idea of our treatment here is to regard maximal consistent subsets of possible arguments as possible worlds. This subset of possible arguments are exactly all those arguments that actually hold true at the corresponding possible world, hence it must be consistent, and also once it is maximal, this subset contains so many possible arguments that they completely cover all aspects of a world’s characteristics, thus uniquely determining the one possible world.

For now, in order to formalize the above intuitive idea, one subtle technical issue remains to be solved with care: we still lack a rigid definition of so-called “consistent” subsets of possible arguments. After all, recall that now we are directly provided with an argument Kripke model, which is never guaranteed to have been actually generated from some epistemic Kripke model; in fact based on Definition 3, we know nothing but attack relations are irreflexive in argument Kripke models. Hence, we have to define a very general notion of “consistency” through very general mathematical tools.

Nevertheless, some sort of hint drawn from our established methodology of the generated argument Kripke model may still prove to be quite enlightening. For a figurative clue let us refer back to Example 1, where the generated argument Kripke model $\mathfrak{N}_{g}$ very similarly resembles the power set lattice of $S$ [1], so apparent an evidence that drives us to naturally think of the bijective correspondence between any element in a set and its principal ultrafilter, which is exactly a subset of the power set. Such a broad orientation via ultrafilters seems quite promising, however as we can confirm, an argument Kripke model does not necessarily form a Boolean algebra, and thus we have to invoke a more generalized mathematical definition, say ultrafilters over a partial order (or preorder, almost equivalently). But then what is the partial order?

Thankfully, recall our previous informal discussion on stronger arguments and weaker arguments in Section 3, there exactly exists a quite handy and natural notion of partial order among arguments, namely by order of argumentation strength. And then immediately, our core philosophical idea bears intimate similarity to the ultrafilter extension construction in modal logic, so that proper filters correspond to “consistent theories” while ultrafilters correspond to “maximal consistent theories”, for readers unfamiliar with this topic, cf. [4] as a rudimentary introduction. Formal definitions go as the following:

Verifying that $\leq$ is reflexive and transitive is obvious. As for the intuitive reading, $U\leq V$ means “argument $U$ is no weaker than argument $V$ in attack power”, just as Definition 6 shows, if and only if any pair of proposition and argument attacked by $V$ also gets attacked by $U$.

Given any fixed partial order (or more generally, fixed preorder) $\leq$ over any fixed nonempty set $\mathbb{W}$, the usual definitions of filters and ultrafilters [16] are presented as below:

Here we have to pay a little more attention, however. Unlike set-theoretic ultrafilters, which always exist for any nonempty set $S$ as subsets of $\mathcal{P}(S)$, for a portion of partial orders or preorders there exist no ultrafilters over them. To give a pictorial though informal depiction, those partial orders without ultrafilters never “diverge permanently” towards totally independent directions, thus so to speak, they are either just linear orders or in a sense much like linear orders. Fortunately, such kind of partial orders matter little to argumentation theory — no need to argue if there is no divergence after all! Therefore, as the following Definition 9, in this paper we can safely concentrate on nontrivial argument Kripke models with ultrafilters.

Through viewing ultrafilters as possible worlds, we at last approach the following Definition 10:

It is easy to see that $S\neq\emptyset$ and that $E(i)$ is an equivalence relation, thus $\mathfrak{M}_{g}$ in Definition 10 is indeed a well-defined epistemic Kripke model. Readers may now wonder, however, that we have yet provided no justification for $E(i)$ or $F(p)$ in Definition 10. As a matter of fact, since possible worlds are constituted of ultrafilters, which are somewhat mathematically factitious after all, it is inevitably tough to give further intuitive explanation solely based on our folk experience. Rather, we shall adopt a little indirect but strictly formal approach, through investigating into a beautiful duality theorem on invariance over the language in Definition 1, so as to demonstrate that our generations in Definitions 5 and 10 are indeed perfectly reasonable. Nevertheless, before we dive into technical subtleties related to this duality theorem in Section 6, we had better first take a look at one vivid practical example presented in the following Section 5, both for acquainting readers with our rather abstract mathematical definitions above and for illustrating the significant practicality of such two-way generation as well as our framework as a whole.

In actuality, the above debate depicted in Example 2 may well continue going on, but to utter a real-time commentary on the debate, at the present moment Alice’s last argument attacks Bob’s, and thus based on our folk principle that “the one who has the last word laughs best”, we can conclude that Alice is tentatively winning the debate. Therefore, we can formalize this debate as the following:

• •

  Two agents $a,b$ denote Alice and Bob, respectively.

• •

  $A_{1}$, $B$, $A_{2}$ denote the three consecutive arguments by Alice and Bob, respectively, as we have indexed beside the text above in Example 2. Our current possible argument is the last argument, namely $A_{2}$.

• •

  Proposition $p$ denotes that “there exists considerable unemployment.” Argument $B$ attacks argument $A_{1}$ with respect to proposition $p$.

• •

  Proposition $q$ denotes that “considering aging, unemployment is not a serious problem”. Argument $A_{2}$ attacks argument $B$ with respect to proposition $q$.

• •

  At present, arguments $A_{1}$ and $A_{2}$ are of course available to Alice herself. Nonetheless, argument $B$ is not available to Bob now (although it used to be), because it is losing to the current argument $A_{2}$ which is attacking it and hence Bob has to give it up.

Then, our current argument Kripke model $\mathfrak{N}$ should look like the following:

Based on model $\mathfrak{N}$, the preorder $\leq$ in Definition 6 can be represented as the following Hasse diagram, namely $A_{2}\leq A_{1}$ and $B\leq A_{1}$:

And so obviously there exist two ultrafilters, namely $\{A_{1},A_{2}\}$ and $\{A_{1},B\}$.

Before finalizing our construction of the generated epistemic Kripke model $\mathfrak{M}_{g}$, let us pause for a while and genuinely think about what we are actually doing now and why we are doing it. Surely, we desire a familiar, tidy and expressive epistemic Kripke model that can aid us to reason about the current scenario, as epistemic logic has been thoroughly studied over the decades and has proved to be a quite accurate and productive depiction of agents’ knowledge, especially helpful when dealing with intricate situations such as the muddy children puzzle and the Byzantine generals problem [9]. Nonetheless, there remains one fatal concern. Indeed, the common method to obtain epistemic Kripke models in reality still relies largely on human intuition and so is rather obscure, a deficiency which in fact greatly impedes such kind of logic approach to knowledge presentation from being universally put into practical application [18]. For instance, just under our current setting of the above Example 2, we have already formalized the argument Kripke model $\mathfrak{N}$ with ease, simply as a literal transcription of the debate per se, however, the task of generating the epistemic Kripke model for Alice and Bob is far from obvious or even natural. After all, in the realistically physical world, it is solely Alice and Bob’s extrinsic debate that is plainly accessible to other audience, whereas their intrinsic knowledge could be indirectly inferred but never directly observed.

The above analysis accounts for exactly why we are about to compute the generated epistemic Kripke model $\mathfrak{M}_{g}$ from our original argument Kripke model $\mathfrak{N}$, and also illuminates one of the major motivations as well as applications of our dual framework presented in this paper. Hence as defined in the previous Section 4, the generated epistemic Kripke model $\mathfrak{M}_{g}$ from the argument Kripke model $\mathfrak{N}$ together with the current possible argument $A_{2}$ is calculated to be the following:

It can be noted that $\mathfrak{M}_{g}$ contains two possible worlds, namely the two ultrafilters $\{A_{1},A_{2}\}$ and $\{A_{1},B\}$. Thus which one should be our current possible world? The current possible world should of course include the current possible argument $A_{2}$, and so it must be the ultrafilter $\{A_{1},A_{2}\}$, where both $p$ and $q$ really hold. Now as $\mathfrak{M}_{g}$ also shows, Alice actually knows the fact $q$ but Bob does not, since he also considers another world (i.e. ultrafilter) $\{A_{1},B\}$ possible, where $\neg q$ holds instead. Hence in a word, our resulting epistemic Kripke model $\mathfrak{M}_{g}$, albeit automatically generated from the argument Kripke model $\mathfrak{N}$, indeed conforms pretty well with what a human being would intuitively deduce from the argumentation process between Alice and Bob in Example 2.

Admittedly, in this specific simple example, $\mathfrak{M}_{g}$ looks so obvious just by our intuition, and some of the readers might still be wondering why we essentially need such an indirect process to generate $\mathfrak{M}_{g}$ rather than try to directly construct the current epistemic Kripke model. But here the main point is, no matter how simple or how complex $\mathfrak{M}_{g}$ may actually be, to say the very least, any straightforward clues about what the epistemic Kripke model is like can be scarcely found either in the real physical world, or in the narration of some story happened in real world. For instance considering the muddy children puzzle, an ordinary solution would commonly start with a cubic epistemic Kripke model which is almost taken for granted, although this puzzle’s literal text — namely some English sentences describing the background settings of the puzzle followed by the conversation between the parent and the children — actually shows no direct hint for any cubic structure at all. As a matter of fact, according to David Lewis’ modal realism [17], all possible worlds in the epistemic Kripke model are equally real, which could suggest that since we are at our current world, we should not be fully capable of observing other possible worlds as directly as the current world, thus not for the whole epistemic Kripke model, either. Therefore in argumentation theory as well as other fields of application, quite substantially, the intangible epistemic Kripke model is always generated from other concrete observables, such as the argument Kripke model. Even though in relatively easy cases like Example 2 and the muddy children puzzle, the epistemic Kripke model seems fairly apparent, nonetheless, human’s intuition may fail to function well when confronted with much complicated circumstances, and moreover importantly, such vague concept of human intuition possesses little reference value for AI systems to be counted as an effective algorithm. Hence, our approach by the dual theory helps to fill up the above gap through automatically generating the appropriate epistemic Kripke model in a systematical way.

As a conclusion to our philosophical analysis over Example 2 in this section, we have carefully demonstrated how to generate the corresponding epistemic Kripke model $\mathfrak{M}_{g}$ given the current argument Kripke model $\mathfrak{N}$. In the next Section 6, we shall then turn our attention to some technical details and in the end prove a duality theorem so as to formally justify our such methodology of two-way generation.

Fix an epistemic Kripke model $\mathfrak{M}=(S,E,F)$ with a possible world $s\in S$, and suppose its corresponding generated argument Kripke model is $\mathfrak{N}_{g}=(\mathbb{W},A,G)$, where we have $\{s\}\in\mathbb{W}$. Assume $\mathfrak{N}_{g}$ is nontrivial. Thus once again, with respect to this fixed argument Kripke model $\mathfrak{N}_{g}$ and this fixed possible argument $\{s\}$, suppose its corresponding generated epistemic Kripke model is $\mathfrak{M}_{gg}=(S^{\prime},E^{\prime},F^{\prime})$. We reasonably expect that $\mathfrak{M}_{gg}$, even though may not be exactly identical to the original $\mathfrak{M}$, must resemble $\mathfrak{M}$ to a certain extent. Nonetheless, in order to pin down such intuition precisely, we firstly have to determine which possible world in $S^{\prime}$ should correspond to the original $s\in S$, namely, represent the currently real world.

Although the definition here is slightly different from ordinary principal ultrafilters over a Boolean algebra [13], readers should feel convinced that the core idea is the same and so we are quite justified for usage of the same name. Then of course, the current possible world in the “generated generated” epistemic Kripke model $\mathfrak{M}_{gg}$ should be $\tau(s)\in S^{\prime}$, namely, the principal ultrafilter of the original current world $s\in S$.

A little more preparation is required: we additionally assume that for any proposition $p\in P$, $\mathfrak{M},s\vDash p$ i.e. $s\in F(p)$. The reason is that an argument Kripke model can only inform us about the relative attack relations between arguments with respect to some proposition $p$, but nothing concerned with their absolute true-hood or falsehood, hence such kind of assumption, or to be called more accurately as convention or protocol, is necessarily needed. Anyhow, this restriction actually does not matter a lot, since due to symmetry between true and false we can always freely choose either $p$ or $\neg p$ as the proper basic proposition so as to meet the assumption’s requirements. We then firstly prove a series of useful lemmata under our current assumption:

Finally we state our main result as the following duality theorem:

It may seem a little pity that according to Theorem 6.1, only agents’ knowledge about atomic propositions (and their negations) is ensured to be preserved from $\mathfrak{M}$ to $\mathfrak{M}_{gg}$, but not necessarily for more complex knowledge like $\mathcal{K}_{i}(p\to q)$ or mutual knowledge like $\mathcal{K}_{i}\mathcal{K}_{j}p$. However, such limitation is exactly due to the fact that in Definition 3 of argument Kripke models, possible arguments can attack each other only with respect to each single atomic proposition, but not with respect to a set of propositions like $\{p,q\}$ or agents’ knowledge like $\mathcal{K}_{j}p$. For simplicity and clarity, we choose to work with this restricted version of definition in the major part of this paper, but shall also briefly discuss possible directions to expand the model and semantics in the following Subsection 7.2 so that we can obtain richer representation and finer-grained resolution.

On the other hand symmetrically, it is also totally feasible to start from an argument Kripke model $\mathfrak{N}=(\mathbb{W},A,G)$ with a possible argument $W\in\mathbb{W}$ and end up with the “generated generated” argument Kripke model $\mathfrak{N}_{gg}$, while a corresponding duality theorem similar to Theorem 6.1 should then be expected under a suitable set of presumptions. Anyhow, this dual generating process from $\mathfrak{N}$ to $\mathfrak{N}_{gg}$ might seem less intuitive or practically useful, and we shall omit detailed analysis here since by duality, the mathematical technique is just very similar.

In sum, this kind of duality theorem formally justifies our two-way generation in Definitions 5 and 10, so that the static, intrinsic epistemic Kripke models and the dynamic, extrinsic argument Kripke models are indeed a dual pair.

This section collects a bunch of miscellaneous discussion on further expansions and applications of our framework presented in this paper.

In this paper, we come up with a dual theory between static knowledge and dynamic argumentation so as to uncover the two concepts’ profound relevance. Particularly, we define in symmetry two types of modal formulae, for knowledge and argumentation respectively, as well as two corresponding types of Kripke models. We then semantically establish the two-way generation method capable of transforming either type of Kripke model to the other type, and also rigidly justify our methodology by showing the syntactic invariance via a dual theorem. Our novel yet convincing approach to combine knowledge and argumentation fundamentally gets rid of any unnatural affectations in other previous attempts, while our simple yet clear framework is supposed to lay out a basic playground for both epistemic logic and argumentation theory.

Moreover, through careful examination on a vivid example of pragmatically utilizing our dual theory, we not only demonstrate in a concrete way how the methodology of two-way generation actually works, but we are also led to the eventual discovery of our framework’s one significantly useful application in practice: realistically, the argument Kripke model is usually much easier to observe and obtain, for example simply by recording the actual verbal debate among the agents, from the source data of which we are then able to generate the corresponding epistemic Kripke model and carry out a real-time analysis over the current debate. What is more important, the above task essentially relies little on human intuition and so is also viable to get mechanically performed by an AI system, who can properly implement modal logic’s decidability as one of its main advantages.

Further, we discuss possible expansions as well as applications of our framework, which may constitute some future research directions. Especially, we claim that in contrary to our common perception, the seemingly “static” knowledge essentially possesses an indispensably dynamic instinct. As Francis Bacon’s famous saying goes: “Knowledge is power”. Such admitted characterization of knowledge also displays a vibrant image to help us intuitively understand knowledge’s dynamic essence, as power can only manifest itself through actual exertion of the power in some specific dynamic event, for instance, by demonstrating your argument power to skillfully defeat your opponent during a debate, while your such power exactly comes out of your knowledge. All the entanglement between statics and dynamics reminds us of Karl Marx’s extremely influential motto: “The philosophers have hitherto only interpreted the world in various ways; the point, however, is to change it”.

Both the authors owe immense gratitude to Teeradaj Racharak (whose nickname is ‘X’) for his teaching and help on argumentation theory. We would also like to thank the anonymous reviewers for their invaluable advice.